Nonrenewable & Renewable Resource Models

Part I: Nonrenewable Resources
1. A three-period problem
a) The expression for the present value of profits from t = 0 to t =2 is
max
q1 ,q2
∫
t =0
t =2
( pt −c ) qt e−rt dt=max
q1 ,q 2
∏
t =0
t=2 pt qt −c qt
( 1+r ) t
b) The expression means sum up over time pt qt −c qt
( 1+r ) t . ( pt −c )qt represents the profits
every period between t = 0 to t = 2.
As a whole, the expression is an objective to choose extraction levels in each period
to maximize the present value of profit over a three-year period.
L= ( pt−c ) qt + λt ( −qt )
∂ L
∂ qt
=0= ( pt−c ) −λt
∂ L
∂ St
=− ̇λ+ rλ=0
c) From (b) we have ∂ L
∂ qt
=0= ( pt−c ) −λt
But pt =10−qt and c=2
Thus,
10−qt −2− λt =0
qt=8−λt
Option value for both q0 ,q1 and q2 is 8−λt .
d) Price in optimum for the periods is
pt =10−qt
p0=10−8+ λt =2+ λt
p1=10−8+ λt=2+ λt
p2=10−8+ λt=2+ λt
The price in optimum remains constant over the periods. This means that the market
conditions over the periods as well as the rate of interest r remain constant over the
period.
e) Recall pt −c−λt =0. Rearranging and substituting known values we obtain;

λt= pt −2
Differentiating,
d λt
d pt
=1
Change of Lagrange multiplier to change in present value is equal to 1. This shows
that a unit change in Lagrange multiplier yields a unit change in present value of the
marginal user cost.
f)
∂ L
∂ St
=− ̇λ+ rλ=0
But we have d λt
d pt
= ̇λt =1
It becomes
−1+r ( pt −2 ) =0
r = 1
pt −2
r = 1
8−qt
Growth rate r is dependent on the demand. The demand on the other hand is
influenced by the interest rate.
2. Finite time-horizon problem in continuous time
a) The expression states that sum up from t=0 to t=T ( p ( t )− q ( t )
2 )q ( t ) e−rt.
( p−q ( t )
2 )q ( t ) is profits in each period. e−rt is the continuous version of the discount
factor.
It is therefore an objective function that is translated as to pick extraction levels in
each period to optimize the present value of profits over T year period.
b) H= ( p− q (t )
q )q ( t )−λ ( t ) q ( t )